Missing Values Relationships

Overview

Lesson 6 of 10 in Rational Number Knowledge Practices. Students use inverse operations, ratio tables, scaling and number lines to find unknown values in equivalent and proportional relationships, then verify their answers using substitution or a second method.

Learning intentions

• WALT use inverse operations to solve for an unknown.
• WALT use ratio tables, scaling and number lines to solve missing values.
• WALT explain the relationship between quantities in an equation or proportion.
• WALT check that a solution is reasonable and equivalent.

Success criteria

• I can identify the relationship between the known values.
• I can find the missing value using an appropriate strategy.
• I can verify my answer by substitution or a second method.
• I can explain my thinking using mathematical language.

Curriculum links

• Number: use multiplicative thinking, fractions, decimals, percentages and proportional relationships.
• Number: recognise and generate equivalent fractions and representations.
• Algebraic thinking: use equations, relationships and unknowns to solve problems.
• Mathematical communication and reasoning: represent, explain, justify and check solutions.

Lesson structure (60 minutes)

1. 0–5 minutes – Hook and retrieval

Open with the hook and retrieval slides. Display: “A taniwha has eaten one number from each equation. How can we find it without guessing?” Students solve two quick problems on mini-whiteboards: (4 \times \square = 28) and (3/5 = \square/10). Invite students to share the operation that undoes multiplication or scaling.

2. 5–15 minutes – Explicit teaching and modelling

Use the modelling slides and a balance diagram to emphasise that an equation remains balanced. Model (3/5=x/20): the denominator has been multiplied by 4, so multiply the numerator by 4, giving (x=12). Show the same relationship on a ratio table and number line.

Model (40%) of (\square=24). Write (0.4 \times \text{whole}=24), then reason backwards using the inverse operation: (24 \div 0.4=60). Ask: “What stays constant? Which operation undoes the one used? How can we check?”

3. 15–25 minutes – Guided missing-value chains

Distribute the guided missing-value worksheet and complete the first two chains together. For example: (2:5=8:\square), then (25%) of (\square=15). Think aloud while identifying the relationship, choosing a strategy and checking by substitution.

Students solve the next two questions in pairs on mini-whiteboards. Pause for discussion: “Could a ratio table, scaling or a number line show the same solution?” Accept verbal or drawn explanations.

4. 25–43 minutes – Collaborative problem stations

In groups of four, students rotate through three teacher-prepared stations, spending about six minutes at each. Use the station instruction slides to introduce the tasks and display the discussion prompts.

• Taniwha number puzzles: complete linked equations and explain the operation used.
• Garden rows: use ratio tables to find missing numbers of plants in equivalent rows.
• Sports statistics: find unknown totals or percentages, such as (30%) of (\square=18).

Each group records one solution and one verification on the worksheet. Roles are solver, recorder, checker and explainer. Encourage students to compare methods rather than simply compare answers.

5. 43–52 minutes – Targeted teaching and challenge

Bring together students who need support for a short teacher-led group using concrete ratio tables, colour-coded steps and balance diagrams. Revisit one example slowly, then ask students to solve a similar problem with a partner.

Other students complete the remaining worksheet questions. Students ready for extension create a problem with two possible representations but only one missing value—for example, a fraction and percentage representation—and swap it with another pair to solve and verify.

6. 52–60 minutes – Plenary and exit ticket

Use the reflection and exit-ticket slides. Ask students to explain which strategy is most useful when the relationship is shown as a fraction, percentage or ratio. Finish with the exit ticket: (0.6=\square/10) and (30%) of (\square=18). Students must show a check for at least one answer.

Resources

• the Missing Values teaching deck
• the guided missing-value worksheet
• Balance diagrams or balance-scale visuals
• Ratio tables and blank number lines
• Equation cards for teacher-prepared station tasks
• Mini-whiteboards, pens and erasers
• Counters or connecting cubes for concrete modelling
• Coloured pencils for colour-coded steps
• Taniwha, garden-row and sports-statistics problem cards
• Exit-ticket slips or exercise books

Assessment

• Listen for students identifying what stays constant and selecting an appropriate inverse operation.
• Check worksheet explanations, ratio tables and verifications during group work; note whether students can transfer between representations.
• Use the exit ticket to identify students needing further support with decimal or percentage relationships in the next lesson.

Differentiation

• Support: provide completed first rows in ratio tables, colour-code corresponding quantities, use concrete materials and offer worked examples.
• Support: provide a targeted teacher group and allow students to explain solutions verbally or with a drawing before recording symbols.
• EAL/SEN: display and rehearse the vocabulary—unknown, missing value, inverse operation, relationship, equation and ratio table—using sentence frames such as “The relationship is…”, “I used… because…”.
• Extension: require students to solve one problem in two different ways and create a new problem with two representations leading to the same unknown.